Unit 1: Convexity Mathematics II Departament de Matemtiques per a - - PowerPoint PPT Presentation

Convex sets Convex and concave functions

Unit 1: Convexity

Mathematics II

Departament de Matemàtiques per a l’Economia i l’Empresa

Academic year 2010/2011

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Index

1

Convex sets

2

Convex and concave functions

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Index

1

Convex sets

2

Convex and concave functions

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Teorema

Theorem If A is a diagonal matrix, then: it is positive definite if all the entries in the main diagonal are > 0. it is negative definite if all the entries in the main diagonal are < 0. positive semi-definite if all the entries in the main diagonal are ≥ 0. negative semi-definite if all the entries in the main diagonal are ≤ 0 indefinite if it contains entries > 0 y < 0 in the main diagonal.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definitions

Definition A square matrix is regular if its determinant is = 0. Definition A square matrix is singular if its determinant is 0.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definitions

Definition The k-th order principal minors of a n × n matrix A are the determinants of the submatrices formed by k rows of A (in

• rder) and the same k columns.

Definition The k-th order leading principal minor of a n × n matrix A is the principal minor formed by the k first rows and the k first columns of A.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method

Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0, then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0, i.e., A1 < 0, A2 > 0, . . . , then A is negative definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method

Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0, then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0, i.e., A1 < 0, A2 > 0, . . . , then A is negative definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method

Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0, then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0, i.e., A1 < 0, A2 > 0, . . . , then A is negative definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method

Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0, then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0, i.e., A1 < 0, A2 > 0, . . . , then A is negative definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method (2)

Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0, then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0, i.e., A1 ≤ 0, A2 ≥ 0, . . . , then A is negative semi-definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method (2)

Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0, then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0, i.e., A1 ≤ 0, A2 ≥ 0, . . . , then A is negative semi-definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method (2)

Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0, then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0, i.e., A1 ≤ 0, A2 ≥ 0, . . . , then A is negative semi-definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Jacobi method (2)

Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0, then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0, i.e., A1 ≤ 0, A2 ≥ 0, . . . , then A is negative semi-definite. In any other case, A is indefinite.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Concepts

Definition If p y q are two points in Rn, p = q, the line that goes through both points can be expressed as the set of points satisfying: x = (1 − λ)p + λq, λ ∈ R If we consider the points that satisfy: x = (1 − λ)p + λq, λ ∈ [0, 1] then we get the points in the segment joining p and q.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Concepts

Definition If p y q are two points in Rn, p = q, the line that goes through both points can be expressed as the set of points satisfying: x = (1 − λ)p + λq, λ ∈ R If we consider the points that satisfy: x = (1 − λ)p + λq, λ ∈ [0, 1] then we get the points in the segment joining p and q.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definition

Definition A set C ⊂ Rn is convex if for any pair of points x, y ∈ C and 0 ≤ λ ≤ 1, then (1 − λ)x + λy ∈ C. This means that C is convex if for any pair of points in C, the points in the segment that joins them are in C, too. The empty set ∅ and sets containing only one point are convex.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definition

Definition A set C ⊂ Rn is convex if for any pair of points x, y ∈ C and 0 ≤ λ ≤ 1, then (1 − λ)x + λy ∈ C. This means that C is convex if for any pair of points in C, the points in the segment that joins them are in C, too. The empty set ∅ and sets containing only one point are convex.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definition

Definition A set C ⊂ Rn is convex if for any pair of points x, y ∈ C and 0 ≤ λ ≤ 1, then (1 − λ)x + λy ∈ C. This means that C is convex if for any pair of points in C, the points in the segment that joins them are in C, too. The empty set ∅ and sets containing only one point are convex.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Hyperplane

Definition An hyperplane in Rn is a subset of the form H = {x ∈ Rn|c1x1 + c2x2 + . . . + cnxn = α} where ci, α ∈ R. If we denote by c the vector whose elements are ci, we can express the previous set as H = {x ∈ Rn|cx = α}. Theorem Hyperplanes are convex sets.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Semispace

Definition A semispace in Rn is a subset of the form S = {x ∈ Rn|c1x1 + c2x2 + . . . + cnxn ≤ α} where ci, α ∈ R. Theorem Semispaces are convex sets. Theorem The intersection of convex sets is a convex set.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Semispace

Definition A semispace in Rn is a subset of the form S = {x ∈ Rn|c1x1 + c2x2 + . . . + cnxn ≤ α} where ci, α ∈ R. Theorem Semispaces are convex sets. Theorem The intersection of convex sets is a convex set.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definition

Definition Let f : D ⊂ Rn → R be a function defined on a convex set D. Then f is convex if for every pair x, y ∈ D and 0 ≤ λ ≤ 1 it holds: f((1 − λ)x + λy) ≤ (1 − λ)f(x) + λf(y) Definition Let f : D ⊂ Rn → R be a function defined on a convex set D. Then f is concave if for every pair x, y ∈ D and 0 ≤ λ ≤ 1 it holds: f((1 − λ)x + λy) ≥ (1 − λ)f(x) + λf(y)

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Definition

Definition Let f : D ⊂ Rn → R be a function defined on a convex set D. Then f is convex if for every pair x, y ∈ D and 0 ≤ λ ≤ 1 it holds: f((1 − λ)x + λy) ≤ (1 − λ)f(x) + λf(y) Definition Let f : D ⊂ Rn → R be a function defined on a convex set D. Then f is concave if for every pair x, y ∈ D and 0 ≤ λ ≤ 1 it holds: f((1 − λ)x + λy) ≥ (1 − λ)f(x) + λf(y)

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Theorem

Theorem Let f : D ⊂ Rn → R be a class C2 function on a convex open set D.

1

If Hf(x) is positive definite (or semi-definite) for every point x ∈ D, f is convex.

2

If Hf(x) is negative definite (or semi-definite) for every point x ∈ D, f is concave.

Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions

Properties of convex/concave functions

Property Let f : D ⊂ Rn → R be a function defined on a convex set D.

4

If f is convex and α ∈ R, the lower level set Dα = {x ∈ D|f(x) ≤ α} is convex.

5

If f es concave and α ∈ R, the upper level set Dα = {x ∈ D|f(x) ≥ α} is convex.

Mathematics II Unit 1: Convexity